Near tip quasi-static crack growth behavior in strain hardening and softening material

Analyzed in this work is a semi-infinite crack that grows slowly in a steady-state. The assumed constitutive relation for the material permits strain hardening and softening as it is damaged in time. Four distinct regions divided angularly are identified for the asymptotic expressions of the quasi-static crack-tip stress field. They refer to material degraded in front of the crack; undergone elastic unloading; reloading of degraded material; and material completely by exhausted in its load carrying capacity.     

The effect of straining mode on void growth

Large strain finite element method is employed to investigate the effect of straining mode on void growth. Axisymmetric cell model embedded with spherical void is controlled by constant triaxiality loading, while plane-stress model containing a circular void is loaded by constant ratio of straining. Elastic-plastic material is used for the matrix in both cases. It is concluded that, besides the known effect of triaxiality, the straining mode which intensifies the plastic concentration around the void is also a void growth stimulator. Experimental results are cited to justify the computation results. This paper is jointly supported by the National Natural Science Foundation of China (19872064), the Chinese Academy of Sciences (KJ951-1-201) and the Laboratory for Nonlinear mechanics of Continuous Media of the Institute of Mechanics     

The size effect on void growth in ductile materials

We have extended the Rice-Tracey model (J. Mech. Phys. Solids 17 (1969) 201) of void growth to account for the void size effect based on the Taylor dislocation model, and have found that small voids tend to grow slower than large voids. For a perfectly plastic solid, the void size effect comes into play through the ratio εl/R₀, where l is the intrinsic material length on the order of microns, ε the remote effective strain, and R₀ the void size. For micron-sized voids and small remote effective strain such that εl/R₀?0.02, the void size influences the void growth rate only at high stress triaxialities. However, for sub-micron-sized voids and relatively large effective strain such that εl/R₀>0.2, the void size has a significant effect on the void growth rate at all levels of stress triaxiality. We have also obtained the asymptotic solutions of void growth rate at high stress triaxialities accounting for the void size effect. For εl/R₀>0.2, the void growth rate scales with the square of mean stress, rather than the exponential function in the Rice-Tracey model (1969). The void size effect in a power-law hardening solid has also been studied.     

A note on the effect of surface energy and void size to void growth

Recent studies revealed that rapid void growth is the dominant failure mechanism in an elasto-plastic solid under high mean tensile stress. This paper studies the effect of the surface energy and void size to the void growth. The models of a thick spherical shell and a thick spherical column in void growth are analyzed and numerically estimated. The main conclusion from this study is that, for typical metals, the surface energy effect is negligible for voids larger than 100 nm in size, but it may become significant when the void size is on the order of 10 nm.     

Influences of microscope size effect of matrix on plastic potential and void growth of porous materials

Based on approximate theoretical analyses on a typical spherical cell containing a spherical microvoid, the influences of matrix materials' microscopic scale on the macroscopic constitutives potential theory of porous material and microvoid growth have been investigated in detail. By assuming that the plastic deformation behavior of matrix materials follows the strain gradient (SG) plastic theory involving the stretch and rotation gradients, the ratio (λ=l/a) of the matrix materials' intrinsic characteristic lengthl to the microvoid radiusa is introduced into the plastic constitutives potential and the void growth law. The present results indicate that, when the radiusa of microvoids is comparable with the intrinsic characteristic lengthl of the matrix materials, the influence of microscopic size effect on neither the constitutive potential nor the microvoid evolution predicted can be ignored. And when the void radiusa is much lager than the intrinsic characteristic lengthl of the matrix materials, the present model can retrogress automatically to the improved Gurson model that takes into account the strain hardening effect of matrix materials. Project supported by the National Natural Science Foundation of China (No. A10102006).     

The dynamic growth of a single void in a viscoplastic material under transient hydrostatic loading

We have examined the problem of the dynamic growth of a single spherical void in an elastic-viscoplastic medium, with a view towards addressing a number of problems that arise during the dynamic failure of metals. Particular attention is paid to inertial, thermal and rate-dependent effects, which have not previously been thoroughly studied in a combined setting. It is shown that the critical stress for unstable growth of the void in the quasistatic case is strongly affected by the thermal softening of the material (in adiabatic calculations). Thermal softening has the effect of lowering the critical stress, and has a stronger influence at high strain hardening exponents. It is shown that the thermally diffusive case for quasistatic void growth in rate-dependent materials is strongly affected by the initial void size, because of the length scale introduced by the thermal diffusion. The effects of inertia are quantified, and it is demonstrated that inertial effects are small in the early stages of void growth and are strongly dependent on the initial size of the void and the rate of loading. Under supercritical loading for the inertial problem, voids of all sizes achieve a constant absolute void growth rate in the long term. Inertia first impedes, but finally promotes dynamic void growth under a subcritical loading. For dynamic void growth, the effect of rate-hardening is to reduce the rate of void growth in comparison to the rate-independent case, and to reduce the final relative void growth achieved.     

The dynamic growth of a void in a plastic material and an application to fracture

The problem of the dynamic growth of a spherical void in applied stress and strain-rate fields is solved approximately using a variational method. The solution is applied to a model of fracture by void growth from brittle or incoherent inclusions. It is shown that the speed of fracture is limited by inertia near the crack tip. Some numerical results are obtained from data for high strength steels.     

Constitutive potential for ductile damaged material and void growth rate

In this paper a general formula for plastic potential including isotropic ductile damage has been presented on the basis of the thermodynamics for irreversible process and intemal variable theory. With this formula the mass conservation law is satisfied and it also contains a series of unknow coefficients which are the function of macro equivalent stress and the average micro equivalent stress and an unknown function which is the function of two generalized forces. The approximate yield surface equation for isotropically damaged materials is developed. Using this equation the void growth rate is calculated for nonlinear material containing voids. The present results are in good agreement with the numerical results given by the cell model.     

The coupled effects of plastic strain gradient and thermal softening on the dynamic growth of voids

This paper examines the combined effects of temperature, strain gradient and inertia on the growth of voids in ductile fracture. A dislocation-based gradient plasticity theory [J. Mech. Phys. Solids 47 (1999) 1239, J. Mech. Phys. Solids 48 (2000) 99] is applied, and temperature effects are incorporated. Since a strong size-dependence is introduced into the dynamic growth of voids through gradient plasticity, a cut-off size is then set by the stress level of the applied loading. Only those voids that are initially larger than the cut-off size can grow rapidly. At the early stages of void growth, the effects of strain gradients greatly increase the stress level. Therefore, thermal softening has a strong effect in lowering the threshold stress for the unstable growth of voids. Once the voids start rapid growth, however, the influence of strain gradients will decrease, and the rate of dynamic void growth predicted by strain gradient plasticity approaches that predicted by classical plasticity theories.     

The analysis of void growth that leads to central bursts during extrusion

Using large deformation finite element analysis together with Gurson's constitutive model, we have studied the behavior of microvoids nucleated at second phase particles during direct axisymmetric extrusion. Two different die-designs were analyzed. Experiments show that the first die-design results in central burst formation while the second gives a solid product free of central bursts. Comparison of the stress fields of the two die-designs provides a possible explanation of how central bursting initiates and why it appears after several steps of multi-step extrusions. The finite element results are in agreement with experimental observation and show that the finite element method can be successfully used to predict the formation of central bursts during extrusion.     

On the mechanism of void growth and the effect of straining mode in ductile materials

Finite element analysis is employed to investigate void growth embedded in elastic–plastic matrix material. Axisymmetric and plane stress conditions are considered. The simulation of void growth in a unit cell model is carried out over a wide range of triaxial tensile stressing or large plastic straining for various strain hardening materials to study the mechanism of void growth in ductile materials. Triaxial tension and large plastic strain encircling around the void are found to be of most importance for driving void growth. The straining mode of incremental loading which favors the necessary strain concentration around void for its growth can be characterized by the vanishing condition of a parameter called “the third invariant of generalized strain rate”. Under this condition, it accentuates the internal strain concentration and the strain energy stored/dissipated within the material layer surrounding the void. Experimental results are cited to justify the effect of this loading parameter.     

Plastic shearing of materials exhibiting strain hardening or strain softening

We consider the plastic shearing of a strain-rate dependent material exhibiting strain hardening or strain softening, subjected to steady shearing. We establish the existence of classical solutions and study the stability of uniform shearing. For materials exhibiting strain hardening or a moderate degree of strain softening we show that, as t , every solution approaches, at specific rates of convergence, uniform shearing; thus shear bands do not form.     

Two mechanisms of ductile fracture: void by void growth versus multiple void interaction

Two distinct mechanisms of crack initiation and advance by void growth have been identified in the literature on the mechanics of ductile fracture. One is the interaction a single void with the crack tip characterizing initiation and the subsequent void by void advance of the tip. This mechanism is represented by the early model of Rice and Johnson and the subsequent more detailed numerical computations of McMeeking and coworkers on a single void interacting with a crack tip. The second mechanism involves the simultaneous interaction of multiple voids on the plane ahead of the crack tip both during initiation and in subsequent crack growth. This mechanism is revealed by models with an embedded fracture process zone, such as those developed by Tvergaard and Hutchinson. While both mechanisms are based on void nucleation, growth and coalescence, the inferences from them with regard to crack growth initiation and growth are quantitatively different. The present paper provides a formulation and numerical analysis of a two-dimensional plane strain model with multiple discrete voids located ahead of a pre-existing crack tip. At initial void volume fractions that are sufficiently low, initiation and growth is approximately represented by the void by void mechanism. At somewhat higher initial void volume fractions, a transition in behavior occurs whereby many voids ahead of the tip grow at comparable rates and their interaction determines initiation toughness and crack growth resistance. The study demonstrates that improvements to be expected in fracture toughness by reducing the population of second phase particles responsible for nucleating voids cannot be understood in terms of trends of one mechanism alone. The transition from one mechanism to the other must be taken into account.     

Damage accumulation and crack growth in bilinear materials with softening: Application of strain energy density theory

A pseudo-elastic damage-accumulation model is developed by application of the strain energy density theory. The three-point bending specimen is analyzed to illustrate the crack growth characteristics according to a linear elastic softening constitutive law that is typical of concrete materials. Damage accumulation is accounted for by the decrease of elastic modulus and fracture toughness. Both of these effects are assessed by means of the strain energy density functions in the elements around a slowly moving crack. The rate of change of the strain energy density factor S with crack growth as expressed by the relation dS/da = constant is shown to describe the failure behavior of concrete. Results are obtained for different loading steps that yield different slopes of lines in an S versus a (crack length) plot. The lines rotate about the common intersect in an anti-clockwise direction as the load steps are increased. The intersect shifts upward according to increase in the specimen size. In this way, the combined interaction of material properties, load steps and specimen geometry and size are easily analyzed in terms of the failure mode or behavior that can change from the very brittle to the ductile involving stable crack growth. An upper limit on specimen or structural size is established beyond which stable crack growth ceases to occur and failure corresponds to unstable crack propagation or catastrophic fracture. The parameters that control the failure mode are the threshold values of the strain energy density function (dW/dV)_c and the strain energy density factor S_c.     

FCC晶体中不均匀变形对空洞长大的影响

通过编制率相关有限元用户子程序,采用一个单胞模型研究了FCC晶体中孔洞在单晶及晶界的长大行为,分析了由于晶体取向及变形失配对孔洞长大和聚合的影响。研究结果表明：孔洞的形状和长大方向与晶体取向密切相关;晶界上孔洞的长大速度大于单晶中孔洞的长大速度;晶粒间的变形失配加速了晶界上孔洞的长大趋势,因而使材料易发生沿晶断裂,随着晶粒间取向因子差异的增加,孔洞越易沿着晶界长大。     

The modified Gurson model accounting for the void size effect

The Gurson model [J. Engrg. Mater. Technol. 99 (1977) 2] has been widely used to study the deformation and failure of metallic materials containing microvoids. The void volume fraction is the only parameter representing voids since the void size does not come into play in the Gurson model. Based on the Taylor dislocation model [Proc. R. Soc. (Lond.) A145 (1934) 362; J. Int. Metals 62 (1938) 307], we extend the Gurson model to account for the void size effect. It is shown that the yield surfaces for micron- and submicron-sized voids are significantly larger than that given by the Gurson model. For a voided, dilating material subject to uniaxial tension, the void size has essentially no effect on the stress–strain curve at small initial void volume fraction. However, as the initial void volume fraction increases, the void size effect may become significant.     

应变软化对复合材料分层尖端场的影响

提出了适用于纤维增强复合材料应变软化分析的多标量连续损伤模型,并将其应用于含分层裂纹的复合材料层板的后屈曲损伤破坏有限元分析,研究了软化参数对损伤场和应力场的影响。计算结果表明:(1)应变软化使分层尖端的应力奇异性降低或消失。(2)应变软化参数影响极限损伤区的大小及其扩展速度,应力跌落使极限损伤加剧。(3)由于损伤的影响,裂尖的能量释放率受软化参数的影响出现波动,已难反映分层特征,可以根据损伤程度来判断分层的扩展。(4)分层上下表面的纤维方向影响损伤形式及其扩展方向。     

Dynamic void growth in rate-sensitive plastic solids

A dynamic void growth model in rate-sensitive plastic materials is derived. The constitutive relation of the matrix is in the overstress form proposed by Perzyna. When the rate of deformation sensitive parameter tends to zero, the Gurson model is retrieved. When the porous material element is under triaxial tension, the Carroll-Holt and Johnson models are retrieved. The normality condition of the plastic rate of deformation to the dynamic loading surface at constant equivalent rates of deformation (with the volumetric part also taken into account) is discussed, and it is shown that the normality rule no longer exists in general. Finally, an approximate expression of the dynamic loading surface that may be convenient for engineering applications is suggested.